Time-dependent problems

For a time-dependent problem — a transition path, a finite horizon, a deterministic change in parameters — pass a strictly increasing time grid as the fourth positional argument:

τs = range(0, 100, length = 50)
result = pdesolve(pde, stategrid, guess, τs)

pdesolve solves backward over this grid: guess is the terminal value at τs[end], and each step solves one implicit time step of $\partial_t v = \text{vt}$ — the same equation, and the same sign convention, as the stationary false transient (see Writing the PDE function). Each solution array gains a trailing time dimension: result.solution.v[.., i] is the value function at time τs[i], and result.residual_norm[i] the corresponding residual. The stationary pseudo-transient controls Δ, scale, minΔ, and maxΔ are not accepted in this mode; the intervals in τs define the implicit time steps. Inner nonlinear-solve controls such as inner_maxiters, inner_abstol, and inner_verbose still apply.

Time-varying equations

If the equation itself depends on time, define the PDE function with a third argument. For example, productivity following a deterministic path:

function pde(state::NamedTuple, u::NamedTuple, t)
    A_t = A * (1 + 0.1 * exp(-0.05 * t))
    # ... same as the stationary function, with A_t in place of A ...
    return (; vt)
end

If the PDE function has only two arguments, the same equation is used at every time on the grid. pdesolve detects the three-argument form from the actual first grid point and terminal time, so a concrete time annotation such as t::Float64 is fine when the supplied time grid contains Float64 values.

Typical uses

Finite-horizon problems: the terminal value is known (e.g. zero, or a bequest value) and the object of interest is the whole path. See Tuckman-Vila.
Transition dynamics: solve the stationary problem first, then use it as the terminal value of a long time grid to trace convergence from an initial condition.
A robust path to a hard stationary solution: when the stationary solve fails from a poor guess, integrating the time-dependent equation over a long horizon is a natural continuation method — it follows the physically meaningful transient.